Or the 2-4-6 game 'reversed', yes. Before Misha's post, I was actually going to try and straighten out the confusion over "logically faultless communication" by showing how it would apply to the 2-4-6 game 'reversed' as an example. I might still, depending.

Although the thing is that I'd say the best way to communicate something like '2-4-6' isn't as one of the simpler concepts in the hierarchy, but as a 'cognitive routine', which is made of a chain (possibly branching) of various simpler concepts that have already been taught.

As Misha said:

For instance, teaching integration by substitution might first involve a simple sequence of examples about identifying when the method is appropriate, then a sequence about choosing the correct substitution, before actually teaching students to solve an integration problem using the method.

(Which doesn't even get into the hidden complexity of all parts 'black boxed' together in the sentence as assumed already taught)

Of course it is easier to explain "logically faultless communication" by showing how it applies to more basic concepts, than to complex concepts that are made up of many of those basic concepts connected together.

The problem is that when you just show the very basic concepts as the AthabascaU module on DI does, people say stuff like this:

And, more generally, all examples given may be used for teaching categorization of objects. How do you teach algorithms (such as multiplication)? How do you teach history and geography? How do you teach calculus? How do you teach scientific method? Not every knowledge can be reduced to questions of form "does X have property Y" taught by presenting series of objects which either are or aren't Y. In the whole presentation there was not a single practically applicable example. Children don't need to go to school to learn what "is longer than" or "not horizontally aligned" means.

(Prase, in comment on first DI post.)

I anticipated this, and had tried to avoid it by injecting a little excitement, like: 'Hey y'all, here's something extremely valuable but complex and non-obvious. It will seem confusing and/or trivial at first, but it really is valuable!'

And in actual fact, looking back, that probably did help, because I got plenty of people going, 'What the hell are you talking about? Give us some meat!' rather than just, 'Huh. Whatever.'

I'm sure there was a much better way possible of achieving the same goal, but what were the chances of me ever finding it without any feedback on actual attempts?